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(* Title: FOLP/ex/Propositional_Int.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1991 University of Cambridge
*)
section \<open>First-Order Logic: propositional examples\<close>
theory Propositional_Int
imports IFOLP
begin
text "commutative laws of & and | "
schematic_goal "?p : P & Q --> Q & P"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
schematic_goal "?p : P | Q --> Q | P"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
text "associative laws of & and | "
schematic_goal "?p : (P & Q) & R --> P & (Q & R)"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
schematic_goal "?p : (P | Q) | R --> P | (Q | R)"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
text "distributive laws of & and | "
schematic_goal "?p : (P & Q) | R --> (P | R) & (Q | R)"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
schematic_goal "?p : (P | R) & (Q | R) --> (P & Q) | R"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
schematic_goal "?p : (P | Q) & R --> (P & R) | (Q & R)"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
schematic_goal "?p : (P & R) | (Q & R) --> (P | Q) & R"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
text "Laws involving implication"
schematic_goal "?p : (P-->R) & (Q-->R) <-> (P|Q --> R)"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
schematic_goal "?p : (P & Q --> R) <-> (P--> (Q-->R))"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
schematic_goal "?p : ((P-->R)-->R) --> ((Q-->R)-->R) --> (P&Q-->R) --> R"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
schematic_goal "?p : ~(P-->R) --> ~(Q-->R) --> ~(P&Q-->R)"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
schematic_goal "?p : (P --> Q & R) <-> (P-->Q) & (P-->R)"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
text "Propositions-as-types"
(*The combinator K*)
schematic_goal "?p : P --> (Q --> P)"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
(*The combinator S*)
schematic_goal "?p : (P-->Q-->R) --> (P-->Q) --> (P-->R)"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
(*Converse is classical*)
schematic_goal "?p : (P-->Q) | (P-->R) --> (P --> Q | R)"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
schematic_goal "?p : (P-->Q) --> (~Q --> ~P)"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
text "Schwichtenberg's examples (via T. Nipkow)"
schematic_goal stab_imp: "?p : (((Q-->R)-->R)-->Q) --> (((P-->Q)-->R)-->R)-->P-->Q"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
schematic_goal stab_to_peirce: "?p : (((P --> R) --> R) --> P) --> (((Q --> R) --> R) --> Q)
--> ((P --> Q) --> P) --> P"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
schematic_goal peirce_imp1: "?p : (((Q --> R) --> Q) --> Q)
--> (((P --> Q) --> R) --> P --> Q) --> P --> Q"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
schematic_goal peirce_imp2: "?p : (((P --> R) --> P) --> P) --> ((P --> Q --> R) --> P) --> P"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
schematic_goal mints: "?p : ((((P --> Q) --> P) --> P) --> Q) --> Q"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
schematic_goal mints_solovev: "?p : (P --> (Q --> R) --> Q) --> ((P --> Q) --> R) --> R"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
schematic_goal tatsuta: "?p : (((P7 --> P1) --> P10) --> P4 --> P5)
--> (((P8 --> P2) --> P9) --> P3 --> P10)
--> (P1 --> P8) --> P6 --> P7
--> (((P3 --> P2) --> P9) --> P4)
--> (P1 --> P3) --> (((P6 --> P1) --> P2) --> P9) --> P5"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
schematic_goal tatsuta1: "?p : (((P8 --> P2) --> P9) --> P3 --> P10)
--> (((P3 --> P2) --> P9) --> P4)
--> (((P6 --> P1) --> P2) --> P9)
--> (((P7 --> P1) --> P10) --> P4 --> P5)
--> (P1 --> P3) --> (P1 --> P8) --> P6 --> P7 --> P5"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
end
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